Discrete Mathematics 2010-2018

Vortragende:


József Beck
Vitaly Bergelson
István Berkes
Yuri Bilu
Pietro Corvaja
Michael Drmota
Clemens Fuchs
Kálmán Gy\H{o}ri
Gerhard Larcher
Harald Niederreiter
János Pintz
Klaus Schmidt[2cm]

Aktuelle Informationen unter:
http://www.math.tugraz.at/Tichy60}

Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This result shows that the distribution of zeros of the Riemann zeta function is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function.

Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions when the associated modulo is not small. Finally, we introduce an equivalence condition analogous to that of Speiser’s for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.

After the talk there will be coffee & cake.

Coauthor: Filiz Yıdız, Dept Math, Hacettepe University, Ankara, Turkey

We define and investigate convexity structures in the sense of Takahashi in T_0$-quasi-metric spaces. We note that several important results about convexity structures in metric spaces can be generalized to the quasi-metric setting.

Furthermore various interesting additional conditions that such convexity structures can satisfy are studied.

We observe that such convexity structures naturally occur in asymmetrically normed real vector spaces and Isbell-convex $T_0$-quasi-metric spaces.